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量子アニーリングによる最適化問題の研究

門脇正史

Published Mar 1, 1999• 0 citations
OpenAlex

Abstract

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state.Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach.The idea is tested by the two models, the transverse Ising model and the traveling salesman problem.Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations.The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method.The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible.We also consider the traveling salesman problem.This model can be described by the Ising spin, so that we also apply the same technique to the transverse Ising model.We solve the time-dependent Schrödinger equation numerically for small size systems with various types of exchange interactions of the Ising model.Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters.We check the case of large-size systems by using the quantum Monte Carlo method.The simulation supports the results of the small-size systems, while the dynamics of the Schrödinger equation and the quantum Monte Carlo method are not the same.We find that the simulated annealing by quantum fluctuations has a better performance than the conventional method for the ground state search of the Ising-spin systems.The calculation of the traveling salesman problem is also performed as A.3 Case of Γ(t) = c/ √ t . . . . . . . . . . . . . . . . . . . . . . . .

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We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state.Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach.The idea is tested by the two models, the transverse Ising model and the traveling salesman problem.Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations.The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method.The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible.We also consider the traveling salesman problem.This model can be described by the Ising spin, so that we also apply the same technique to the transverse Ising model.We solve the time-dependent Schrödinger equation numerically for small size systems with various types of exchange interactions of the Ising model.Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters.We check the case of large-size systems by using the quantum Monte Carlo method.The simulation supports the results of the small-size systems, while the dynamics of the Schrödinger equation and the quantum Monte Carlo method are not the same.We find that the simulated annealing by quantum fluctuations has a better performance than the conventional method for the ground state search of the Ising-spin systems.The calculation of the traveling salesman problem is also performed as A.3 Case of Γ(t) = c/ √ t . . . . . . . . . . . . . . . . . . . . . . . .
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem (TSP). Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. TSP can be described by the Ising spin, so that we also apply the same technique to TSP. We directory solve the time-dependent Schrödinger equation for small-size systems and perform the quantum Monte Carlo simulation for large-size systems. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We also find that the relaxation time is quite short for quantum systems by numerical simulations. We consider this is one of the reasons why the annealing in quantum systems have a better performance of finding the optimal state in comparison with classical systems.
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schr\"odinger equation numerically for small size systems with various exchange interactions. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum annealing leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule.
Convergence conditions for quantum annealing are derived for optimization problems represented by the Ising model of a general form. Quantum fluctuations are introduced as a transverse field and/or transverse ferromagnetic interactions, and the time evolution follows the real-time Schrodinger equation. It is shown that the system stays arbitrarily close to the instantaneous ground state, finally reaching the target optimal state, if the strength of quantum fluctuations decreases sufficiently slowly, in particular inversely proportionally to the power of time in the asymptotic region. This is the same condition as the other implementations of quantum annealing, quantum Monte Carlo and Green's function Monte Carlo simulations, in spite of the essential difference in the type of dynamics. The method of analysis is an application of the adiabatic theorem in conjunction with an estimate of a lower bound of the energy gap based on the recently proposed idea of Somma et. al. for the analysis of classical simulated annealing using a classical-quantum correspondence.
Quantum annealing is a promising tool for solving optimization problems, similar in some ways to the traditional (classical) simulated annealing of Kirkpatrick et al. Simulated annealing takes advantage of thermal fluctuations in order to explore the optimization landscape of the problem at hand, whereas quantum annealing employs quantum fluctuations. Intriguingly, quantum annealing has been proved to be more effective than its classical counterpart in many applications. We illustrate the theory and the practical implementation of both classical and quantum annealing – highlighting the crucial differences between these two methods – by means of results recently obtained in experiments, in simple toy-models, and more challenging combinatorial optimization problems (namely, Random Ising model and Travelling Salesman Problem). The techniques used to implement quantum and classical annealing are either deterministic evolutions, for the simplest models, or Monte Carlo approaches, for harder optimization tasks. We discuss the pro and cons of these approaches and their possible connections to the landscape of the problem addressed.
Quantum annealing (QA) is a generic method for solving optimization problems using fictitious quantum fluctuation. The current device performing QA involves controlling the transverse field; it is classically simulatable by using the standard technique for mapping the quantum spin systems to the classical ones. In this sense, the current system for QA is not powerful despite utilizing quantum fluctuation. Hence, we developed a system with a time-dependent Hamiltonian consisting of a combination of the formulated Ising model and the "driver" Hamiltonian with only quantum fluctuation. In the previous study, for a fully connected spin model, quantum fluctuation can be addressed in a relatively simple way. We proved that the fully connected antiferromagnetic interaction can be transformed into a fluctuating transverse field and is thus classically simulatable at sufficiently low temperatures. Using the fluctuating transverse field, we established several ways to simulate part of the nonstoquastic Hamiltonian on classical computers. We formulated a message-passing algorithm in the present study. This algorithm is capable of assessing the performance of QA with part of the nonstoquastic Hamiltonian having a large number of spins. In other words, we developed a different approach for simulating the nonstoquastic Hamiltonian without using the quantum Monte Carlo technique. Our results were validated by comparison to the results obtained by the replica method.
Sufficient conditions for convergence of quantum annealing are derived for optimization problems represented by the Ising model. Three different types of time evolution are considered; the real-time Schrödinger equation, the path-integral Monte Carlo method and the Green's function Monte Carlo method. It is proved that the system under each dynamics reaches the target solution in the limit of infinite time if the transverse field, representing the strength of quantum fluctuations, decreases inversely proportionally to the power of time in the asymptotic region.